Study of the Flavour Changing Neutral Current Decay mode B Kγγ

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1 Study of the Flavour Changing Neutral Current Decay mode B Kγγ arxiv:hep-ph/010160v 3 Oct 00 S. R. Choudhury a, G. C. Joshi b, Namit Mahajan a and B. H. J. McKellar b a. Department of Physics and Astrophysics, University of Delhi, Delhi , India. b. School of Physics, University of Melbourne, Australia. Abstract We study the neutral current flavour changing rare decay mode B Kγγ within the framework of Standard Model, including the long distance contributions. It is found that these long distance contributions can be of the same order as the short distance contribution. Keywords:Rare B decay PACS:13.5.Hw, Hq 1 Introduction Radiative decays of the B-meson are very useful indices for testing the underlying theories of flavour changing neutral currents (FCNC) since the amplitudes are very sensitive to QCD contributions as well as to possible contributions from loops with supersymmetric partner particles. Of the radiative modes, the decay mode B X s γ has been measured experimentally and has also been the subject of extensive theoretical investigations. The advent of the B-factories will probably make possible measurement of radiative decay modes with two photons. Hence the basic two photon amplitude b sγγ has also been studied starting with the work of Lin, Liu and Yao [1], and pursued further in references [], [3]. E mail : src@ducos.ernet.in E mail: joshi@tauon.ph.unimelb.edu.au E mail : nm@ducos.ernet.in, nmahajan@physics.du.ac.in E mail:b.mckellar@physics.unimelb.edu.au 1

2 The b sγγ amplitude falls naturally into two categories; an irreducible contribution and a reducible one where the second photon is attached to the external quark lines of the b sγ amplitude. The irreducible part can be estimated through the basic triangle graph. Evaluating the reducible part at the quark level then presents no problem. For the process B X s γγ it is appropriate to the consider the amplitude as dominated by the quark level amplitude. However, when we consider an exclusive channel B Mγγ with a specific meson M, it is more appropriate to consider the reducible contributions as arising out of emission of the second photon from the external hadron legs of the amplitude B Mγ rather than restricting oneself to quark level considerations and thereby completly neglecting the spectator contributions in B and M. The amplitude B Kγγ then stands out since the basic one photon amplitude B Kγ vanishes for real photons and we may expect the irreducible part of the amplitude to dominate. There will of course be the usual long distance contributions, the most important being the process B K γ followed by decay of K, and B Kη c followed by two photon decay of the η c. The corresponding η contribution will be small owing to the small η coupling to cc but we include it also for completeness. The B Kη amplitude however is anomalously high, probably because of the gluon fusion mechanism. Since η has a sizeable branching ratio into γ, the η contribution to B Kγγ will prove to be the largest resonance contribution. This note reports an estimate of the branching ratio for the process B Kγγ taking account of the irreducible contribution as well as all significant resonance contributions. The earliest investigations of this process restricted themselves to the η c contribution [4]; subsequently an estimate was made of a possible contribution from the resonance η with the η arising through gluon fusion from the basic process b sgg and subsequently decaying into two photons [5]. The η resonance however is very narrow and its contribution will show up as a distinct narrow peak in the γγ invariant mass spectrum. This can experimentally be easily separated and we could restrict our analysis by excluding this resonance, but we include it for completeness. There are of course interference terms from each pair of terms. Unfortunately the only interference term whose sign we can determine is that between the irreducible term and the η c resonance contribution. The other signs are un-

3 known. We quote the largest and the smallest branching ratio, and give enough information for the reader to construct the other possibilities. Irreducible triangle contributions These arise from triangle diagrams with the photon being emitted by the quark loop. The effective Hamiltonian is [1] with H eff = G F V ts V tb O 1 = ( s i c j ) V A ( c j b i ) V A, O = ( s i c i ) V A ( c j b j ) V A, O 3 = ( s i b i ) V A ( q j q j ) V A, O 4 = ( s i b j ) V A ( q j q i ) V A, q q i C i (µ)o i (µ), (1) O 5 = ( s i b i ) V A ( q j q j ) V +A, () O 6 = ( s i b j ) V A ( q j q i ) V +A, O 7 = e 16π s iσ µν (m s P L + m b P R )b i F µν, and O 8 = g 16π s iσ µν (m s P L + m b P R )Tij a b jg a µν. The invariant amplitude corresponding to quark level transition b sγγ (with an incoming b line and an outgoing s line and the two photons being emitted by the quark loop) is M b s = [ 16 αgf V 9π tsv tb + ιb(m s K(m s )P L + m b K(m b )P R)T µν q q ] { ū(p s ) A q J(m q )γρ P L R µνρ (3) q + C( m s L(m s )P L + m b L(m b )P R)ǫ µναβ k α 1 kβ } u(p b )ǫ µ (k 1 )ǫ ν (k ), 3

4 where R µνρ = k 1ν ǫ µρσλ k1 σ kλ k µǫ νρσλ k1 σ kλ + (k 1.k )ǫ µνρσ (k k 1 ) σ, T µν = k µ k 1ν (k 1.k )g µν, A u = 3(C 3 C 5 ) + (C 4 C 6 ), A d = 1 4 [3(C 3 C 5 ) + (C 4 C 6 )], (4) A c = 3(C 1 + C 3 C 5 ) + (C + C 4 C 6 ), A s = A b = 1 4 [3(C 1 + C 3 C 5 ) + (C + C 4 C 6 )], and B = C = 1 4 (3C 6 + C 5 ). To get to M(B Kγγ) from the quark level amplitude (M(b sγγ)), we should replace s Γ b by K Γ B for any Dirac bilinear Γ. Recall that s sγ ρ b b gives a factor ū s γ ρ u b and similarly for sγ ρ γ 5 b, sγ 5 b, and sb, and also that K Γ B is zero for Γ = γ 5, γ ρ γ 5. In this way we get M irred (B Kγγ) = ( 16 αgf V 9π tsv tb )[ 1 K sγρ b B A q J(m q)r µνρ + 1 K sb B {ιb(m s K(m s) + m b K(m b))t µν (5) + C( m s L(m s ) + m bl(m b ))ǫ µναβk α 1 kβ In the above expressions we introduced the functions q }] ǫ µ (k 1 )ǫ ν (k ) J(m ) = I 11 (m ), K(m ) = 4I 11 (m ) I 00 (m ), L(m ) = I 00 (m ), where I pq (m 1 ) = 1 x x p y q dy 0 0 m (k 1.k )xy ιǫ (6) We use the following parametrization for the matrix elements of the quark vector current: K sγ µ b B = ( (p B + p K ) µ m B m ) K q q µ F1 BK (q ) (7) + ( m B m ) K q µ F0 BK (q ), q 4

5 with q = p b p K = k 1 + k. It then follows that K sb B = (m b m s ) 1 [q µ K sγ µ b B ] (8) Also observe that = (m b m s ) 1 (m B m K)F BK 0 (q ). q ρ R µνρ = (k 1 + k ) ρ R µνρ (9) = (k 1.k )ǫ µνρσ (k ρ 1k σ kρ k σ 1 ). For the form factors appearing above, we use the explicit numerical dependence of F(q ) on q given by Cheng et al [6]. 3 Resonance contributions 3.1 The η c resonance The η c contribution to the deacy process comes via the t-channel decay B Kη c, with then η c decaying into two photons. Since this has to be added to other contributions, we need to be careful about the sign of the term. The T-matrix element for this process can be written as ı ı Kγγ T B = Kη c ıh B q m η c + ım ηc Γ η γγ ıh η c c. (10) total Noting that in the lowest order, for any states X, Y we get X T Y = X H Y, Kγγ T B = Kη c T B γγ T η c q m η c + ım ηc Γ η, (11) c total The amplitude γγ T η c is parametrized as [3] γγ T η c = ıb ηc ǫ µναβ ǫ 1 µ ǫ ν k 1α k β. (1) We can determine B ηc from the η c γγ decay rate: ( 1 Γ(η c γγ) = ) 1 d 3 k 1 d 3 k m η (π) 3 k1 0 (π) 3 k 0 (π) δ 4 (k η k 1 k ) γγ T η c. (13) 5

6 We have and so spins γγ T η c = B ηc m η c, Γ(η c γγ) = B η m 3 η c. 16π Next we need the B Kη c amplitude Kη c T B = Kη c H eff B. The relevant piece of H eff for this matrix element is H eff = G ] F V cb V cs [C ( cb) V A ( sc) V A + C 1 ( c i b j ) V A ( s j c i ) V A = G ] F V cb V [C 1 ( cc) V A ( sb) V A + C ( c i c j ) V A ( s j b i ) V A. cs (14) The second expression is obtained by Fierz transforming the first one. Thus H eff = G F V cb V cs (C 1 + C 3 )( cc) V A( sb) V A. (15) Between B and K only the V part of ( sb) V A contributes whereas for the c-loop with two photons only the A part of ( cc) V A is relevant. Hence, Using factorization we can write Define We thus get H eff = G F V cb V cs (C 1 + C 3 )( cγ µγ 5 c)( sγ µ b). (16) Kη c T B = G F V cb V cs(c 1 + C 3 ) K sγµ b B η c cγ µ γ 5 c 0. (17) 0 A c µ η c ıf ηc q µ A c µ = cγ µ γ 5 c q µ K sγ µ b B = F 0 (m η c )(m B m K ) Kη c T B = ı G F V tb V ts f η c F 0 (m η c )(m B m K )(C 1 + C 3 ) (18) where a dipole form of the form factor F 0 (m η c ) is used. 6

7 Now we fix the relative sign between B ηc and f ηc. This can be easily done through the anomaly equation [7] γγ T η c = ǫ 1µ (k 1 )ǫ ν (k )Γ µν (19) = ǫ 1µ (k 1 )ǫ ν (k ) ıe D ǫ π µναβ k f 1k α β ηc Comparing with our definition of B ηc in eq.(1), we identify B ηc = e D 4π f ηc, and see that the relative sign is positive. However, we emphasise that we do not use this theoretical result to determine the numerical value of either B ηc or f ηc, but merely use it to fix the relative sign. The total contribution due to η c resonances is thus M ηc G F = B ηc f ηc V tb V ts (C 1 + C 3 )F 0(m η c )(m B m K ) (0) ǫ 1µ (k 1 )ǫ ν (k )ǫ µναβ k1 α 1 kβ q m η c + ım ηc Γ η. c total 3. The η contribution Analogous to η c, the η-resonance will also contribute to the decay amplitude because of its coupling to c c channel. This contribution, M η,has exactly the same form as eq.(0) with the parameters B ηc, f ηc, F 0 (m η c ), m ηc and Γ η c total being replaced by their η-counterparts. However in this case we cannot define the relative sign of the η and any of the other components of the amplitude. 3.3 The η contribution Like η c and η, the η can also contribute via the effective Hamiltonian, eq.(15). However, unlike η c and η, the η has a very strong coupling to a two gluon state. Theoretical models for η production in B-decay via gg-states exist both when the gluons arise from a basic b sgg process [8] and also when the spectator quark in B-meson emits a gluon to combine with a basic b sg process [4, 5]. These estimates have considerable theoretical uncertainities. 7

8 Fortunately, experimental data on B Kη exists, which is sufficient for our purpose since the invariant mass of the two photons will be strongly peaked at m η. Calling the coupling of Bi K i η (i = +, 0) as F(B i K i η ), the η contribution to our amplitude can be written as M η = B η F(B i K i η )ǫ 1µ (k 1 )ǫ ν (k )ǫ µναβ k α 1 kβ 1 q m η + ım η Γη total (1) where B η is defined as in eq.(1) with η c η. Also, F(B i K i η ) is related to the decay rate Γ(B i K i η ) as: Γ(B i K i η ) = 1 F(B i K i η ) m λ 1 (1, K 16πm B m B, m η m ) () B Once again the relative sign of the contribution is unknown. 3.4 K resonance contribution In this case we have the following situation: followed by and the process with k 1 k. B i (p B ) K i + γ(k 1 ), i = +, 0 K i K i + γ(k ), The effective Hamiltonian for the first vertex is H eff = 4 G ( ) F V cb VcsC emb 7 s 16π L σ µν b R F µν (3) and the T-matrix element T i for the process K i K i + γ is T i = g i K Kγ ǫ δ(p B k 1 )ǫ β (k )k α (p B k 1 ) λ ǫ αβλδ (4) with g i K Kγ to be determined from the decay rate of K i K i +γ. Summing over the final spins and averaging over the initial spins we get 1 3 T i = 1 3 gi (m K m K ), (5) 8

9 so g i is determined from The matrix element of is Γ i = gi 96π O 7 = ( m K m ) 3 K. (6) m K ( ) emb s 16π L σ µν b R F µν ( ) K i (q) O 7 B i emb (p + q) = ı F i [ǫ α (p)p β ǫ β (p)p α ] (7) 3π ǫ σ (q) [ıǫ αβστ q τ g σα q β ] with F i to be determined from the radiative decay B i K i γ. With these definitions the complete T-matrix element can be written as Kγγ T B = M K (8) [ ] = T µν (k 1, k ) + (µ ν, k 1 k ) ǫ µ(k 1 )ǫ ν(k ) with ( T µν emb g i F i ) (k 1, k ) = 4 G F V 16π cb Vcs C 7ǫ ανγδ k α (p B k 1 ) γ k 1β (9) [ ( g δσ (p B k 1 ) δ (p B k 1 ) σ m K ) (p B k 1 ) m K + ım K [ ΓK total ıǫ µβ σ τ (p B k 1 ) τ (g µσ (p B k 1 ) β g β σ (p B k 1 ) µ ) In the last expression we have used the antisymmetry of F µν to write the terms in this particular form. Also note that the sign of this contribution can not be unambiguously fixed as was donefor the η c contribution. ] ]. 4 Results The total contribution to the process B Kγγ is M tot = M irred + M ηc + ξ K M K + ξ η M η + ξ η M η (30) 9

10 where ξ α = ±1 are sign parameters, introduced because the signs of these terms are unknown. The total decay rate is given by ( ) dγ d 1 = sγγ[( 1 s γγ s γγ 51m B π 3 m + m ) K B m 4m ]1 K π B m dθ sin θ M tot 0 B (31) where s γγ is the C.M. energy of the two photons while θ is the angle which the decaying B-meson makes with one of the two photons in the γγ C.M. frame. The numerical values of various parameters used in our calculation are listed in Appendix. Figure 1 shows the spectrum of our results given as a function of the invariant mass of the two photons, for the case of neutral B meson decay. The figure shows the expected resonance peaks in the γγ spectrum at the positions of the η, η, and η c, and the K resonance contribution is spread out across this projection of the Dalitz plot. We have chosen the sign parameters ξ α = +1 in all cases in the graph. Figure shows a comprison between the results when the interference terms are included and when omitted. The various contributions to the branching ratio are given in Table 1, with the maximum branching ratio (ξ α = +1 for all α), and the minimum branching ratio (ξ α = 1 for all α). We see that we predict the branching ratio in the range Br(B Kγγ) (3) and that the largest single contribution to the branching ratio comes from the η resonance term. In fact the resonance terms (neglecting interference) are more than three times the ireducible term. In an attempt to enhance the relative contribution of the irreducible term we have calculated the partial branching ratio with a cut on s γγ > m ηc +Γ ηc 3.0GeV. In this partial branching ratio the irreducible term does dominate and is about twice the K contribution. We could further reduce the nonirreducible background by introducing a further cut in the Dalitz plot to stay above the K peak in s Kγ, but because of the already small branching ratio, we do not suggest this additional cut. For completeness, we also quote the analogous individual contributions to the branching ratio for the corresponding charged decay mode (B + K + γγ) in 10

11 Table 1: Contributions to the B 0 K 0 γγ branching ratio the total branching ratio and with a cut on s γγ Contribution Branching ratio With cut Resonance η c η η K Irreducible Interference η c I η I ± ± 0.3 η c K η K ± ± K I ± 1.18 ± Maximum BR Minimum BR Table. Comparing the the two branching ratios for the neutral and charged deacy modes, one finds that although the η contribution is larger in the case of the charged mode, the overall branching fractions are almost identical, whether one considers the complete branching ratio, or just the part above the cut. This is because of the fact that the larger branching fraction of the charged mode B + K + γ gets compensated by the smaller value of the branching ratio for the mode K + K + γ. From the tables above, it is clear that above the imposed cut, the η does not contribute significantly. But the same is not found to be true for η. In particular, the η Irredicible interference term is quite large above the cut when compared with the total contribution of this particular term. At first sight this is surprising because the cut is well away from the resonance. However the interference term is dominated by the real part of the resonant amplitude, which decreases only like (s m η ) 1 away from the resonance. 11

12 Table : Contributions to the B + K + γγ branching ratio the total branching ratio and with a cut on s γγ Contribution Branching ratio With cut Resonance η c η η K Irreducible Interference η c I η I ± 0.46 ± 0.93 η c K η K ± ± K I ± ± Maximum BR Minimum BR However the remaining factors in the differential rate grow with s, and the interference term acquires a non-negligible contribution above the cut. The value obtained for Br(B Kγγ), is much much higher than the corresponding branching ratio for the inclusive process B X s γγ, where the total branching ratio is quoted to be ( ) 10 7 [3]. Even the exclusive branching ratio with much of the long distance resonance effects eliminated by the cut in s γγ we have suggested could exceed these values. However, it is known [9] that the inclusive rate calculation based on quark transition picture is not valid when MX becomes low, which is true in our case. Thus we should not be too surprised by our larger result for just one part of the inclusive branching ratio. It is worth mentioning at this point that if instead of using the explicit numerical dependence of B K form factors, following Chen etal[10] we assume F0 BK (0) = F1 BK (0) = and F0 BK (q ) = F 0 BK (0) with M ) pole = 6.65 GeV (1 q [11], we get an even higher value for the branching ratio ( < M pole 1

13 BR < ). Observation of this decay experimentally thus is expected to provide an interesting test of the underlying theory leading to this value. In particular the branching ratio above a cut which removes the η, η and η c contributions will be a good test of the magnitude of the irreducible and will thus be sensitive to the effects of high mass intermediate states, even to physics beyond the standard model. Acknowledgements NM would like to thank the University Grants Commission, India, for a fellowship. SRC would like to acknowledge support from DST, Government of India, under the SERC scheme. This work was supported in part by the Australian Research Council. Appendix We give the input parameters used in the numerical calculations. C 1 C C 3 C 4 C 5 C 6 C 7 C Table 3: The approximate values of C is at µ = m b m b = 4.8 GeV m c = 1.5 GeV m t = 175 GeV m s = 0.15 GeV m u = m d = 0 m B 0 = 5.79 GeV Γ B0 total = GeV m B + = GeV Γ B+ total = GeV m K 0 = GeV m ηc = 3 GeV Γ η c total = GeV m K + = GeV B ηc = GeV 1 f ηc = 0.35 GeV 13

14 m η = GeV B η = GeV 1 Γ η total = GeV f η = GeV m η = GeV Γ η total = GeV Br(B 0 K 0 η ) = Br(B + K + η ) = m K 0 = GeV g K 0 Kγ = GeV 1 Γ K 0 total = 50.5 MeV F i = 0.9 m K + = GeV g K + Kγ = GeV 1 Γ K + total = 50.8 MeV We follow the Wolfenstein parametrization of the CKM matrix with A = 0.8 λ = 0. η = 0.34 V tb 1 V ts = Aλ V cb = Aλ V cs = 1 λ References [1] G. L. Lin, J. Liu and Y. P. Yao, Phys. Rev. Lett. 64, 1498 (1990); G. L. Lin, J. Liu and Y. P. Yao, Phys. Rev. D 4, 314 (1990). [] C. V. Chang, G. Lin and Y. P. Yao, Phys. Lett. B 415, 395 (1997) [arxiv:hep-ph/ ]. [3] L. Reina, G. Riccardi and A. Soni, Phys. Lett. B 396, 31 (1997) [arxiv:hep-ph/961387]; Phys. Rev. D 56, 5805 (1997) [arxiv:hep-ph/970653]. [4] M. R. Ahmady, E. Kou and A. Sugamoto, Phys. Rev. D 57, 1997 (1998). [5] M. R. Ahmady, Phys. Rev. D 61, (000) [arxvi:hep-ph/990877]. [6] H. Y. Cheng, C. Y. Cheung and C. W. Hwang, Phys. Rev. D 55, 1559 (1997) [arxiv:hep-ph/960733]. [7] T. P. Cheng and L. F. Li, Gauge theory of elementary particle physics, Clarendon Press, Oxford,

15 [8] A. Ali, J. Chay, C. Greub and P. Ko, Phys. Lett. B 44, 161 (1998) [arxiv:hep-ph/97137]. [9] A. F. Falk, M. B. Wise and I. Dunietz, Phys. Rev. D 51, 1183 (1995) [arxiv:hep-ph/ ]. [10] Y. H. Chen, H. Y. Cheng, B. Tseng and K. C. Yang, Phys. Rev. D 60, (1999) [arxiv:hep-ph/ ]. [11] H. Fusaoka and Y. Koide, Phys. Rev. D 57, 3986 (1998) [arxiv:hepph/97101]. 15

16 ¹¼ ¹¼ ¹¼ Ö Ô ¹¼ ¼ ¾ ¹¼ ¼ ¼ ¼º ½ ½º ¾ Ô ¾º º º ε ¼º½ ¼º¼½ ¼º¼¼½ ¼º¼¼¼½ Ö Ô ¼ ¼ ¼º ½ ½º ¾ Ô ¾º º º ε Figure 1: Our result for the spectrum of B 0 K 0 γγ (above). The same results plotted with logscale on y-axis (below). The parameters used are listed in the appendix. In there figures we have chosen the signs of the amplidudes so that all of the interference terms add constructively, so the figure corresponds to the 16branching ratio of

17 ¼º½ ¼º¼½ ³Ï Ø ÁÒØ Ö Ö Ò Ø ÖÑ ³ ³Ï Ø ÓÙØ ÁÒØ Ö Ö Ò Ø ÖÑ ³ ¼º¼¼½ ¼º¼¼¼½ Ö Ô ¼ ¼ ¼º ½ ½º ¾ ¾º º º Ô Îµ Figure : The spectrum of B 0 K 0 γγ with and without interference terms. The relative signs are chosen to be all positive in the total amplitude. 17